9 Human capital and Growth (Lucas, 1988)
9.1 Introduction
In contrast to the models presented so far, Lucas (1988) assumes that there are two types of assets endogenously accumulated in the economy, physical capital and human capital. The idea is very simplistic and says that in addition to producing, for instance, more infrastructure we also produce better (or more) educated workers. The better educated workers, then, produce more, while using the same amount of labor. Therefore, the labor productivity increases, and this, together with the capital accumulation, may enable long run growth.
It is worth emphasizing that the biggest difference between Romer (1986) and Lucas (1988) models is that the latter endogenizes the process of labor productivity growth through human capital accumulation, when the former thinks of spillover effects.
If presented in one sector form, the final good production side and the asset accumulation processes of Lucas (1988) model can be written as
\[ Y=A K^{\alpha} H^{1-\alpha}=C+I_{K}+I_{H} \]
where \(H\) is the human capital input, \(I_{K}\) and \(I_{H}\) are the investments for physical capital and human capital accumulation, i.e.,
\[ \begin{aligned} I_{K} & =\dot{K}+\delta_{K} K \\ I_{H} & =\dot{H}+\delta_{H} H \end{aligned} \]
where \(\delta_{K}\) is the depreciation rate physical capital and \(\delta_{H}\) is the depreciation rate of human capital. Given that we consider an equilibrium where both assets are accumulated, the returns to both assets should be equal. For the current exercise let \(\delta_{K}=\delta_{H} \equiv \delta\). This would imply that,
\[ \begin{aligned} \frac{\partial Y}{\partial K}-\delta & =\frac{\partial Y}{\partial H}-\delta \Rightarrow \\ \alpha \frac{Y}{K} & =(1-\alpha) \frac{Y}{H} \Rightarrow \\ H & =\frac{1-\alpha}{\alpha} K \Rightarrow \\ Y & =\left[A\left(\frac{1-\alpha}{\alpha}\right)^{1-\alpha}\right] K \end{aligned} \]
Thus, in terms of structure, the ideas behind the Romer (1986) and Lucas (1988) models are quite similar. Both end up having an aggregate production function with non-decreasing returns to scale, and in particular linear in capital. However, Romer (1986) introduces internal spillover while Lucas (1988) considers external effects through human capital accumulation.
This one sector model was a simple representation of Lucas (1988). The model with corresponding assumptions is the following.
9.2 the Model
This is a two-sector model of growth, where the physical capital is still produced with the same technology as the consumption good, but human capital is produced with a different technology. Human capital is the essential input for the production of new human capital. The motivation for this is that the human capital of one generation is an important factor in affecting the formation of human capital of the later generations. If the production of human capital is within a household, that would be the human capital “embodied” in the parents. If its production is through formal education, then that would be the human capital of the teachers with their methodologies (e.g., books). The accumulation (production) of human capital \(H\) follows a law of motion:
\[ \dot{H}=B H_{H}-\delta_{H} H, \]
where \(H_{H}\) is the human capital used for its own production. Every unit of human capital produces \(B>0\) new units of human capital. This stock depreciates at a rate \(\delta_{H}>0\) (e.g., due to “aging” for instance).
There are no diminishing returns to the production of human capital with this type of production function for the human capital. The non-decreasing returns to the production of human capital will be the engine of long-run growth in this model. The increasing stock of human capital drives the accumulation of physical capital and the economy grows indefinitely. If, instead, the production of human capital had decreasing returns to its input, this model would have the same predictions as the Solow-Swan model and it would not be able explain growth in the long-run.
The production of final output combines physical capital stock and human capital \(H_{Y}\), i.e., \(Y=A K^{\alpha} H_{Y}^{1-\alpha}\), where \(H_{Y}\) is the human capital used in production of final good. Standard neoclassical assumptions apply.
The representative household chooses its consumption path, the assets (physical and human capital) in the next period and the allocation of its human capital input between final good and human capital production, in order to maximize its lifetime utility \(U=\int_{0}^{\infty} u(C) e^{-\rho t} d t\), subject to standard budget constraint and the law of motion of human capital, where \(u(C)=\frac{C^{1-\theta}-1}{1-\theta}\).
Define the fraction of human capital used in the production of final output as \(u \equiv \frac{H_{Y}}{H}\) and its complement \(1-u=\frac{H_H}{H}\). There are no externalities involved in the input and output markets. By the first welfare theorem it is known that the competitive equilibrium will achieve the first-best allocations. The second welfare theorem implies that one can directly solve for the optimal allocations, as there are prices that will support the competitive equilibrium that achieves such intratemporal and intertemporal allocations.
The intertemporal allocation problem has two controls, consumption and allocation of human capital in the two sectors of production that compete for it. There are two state variables, human and physical capital. Physical capital accumulation requires the saving of output (consumption choices), while the human capital accumulation requires investments in terms of real resources that is to say some human capital needs to be driven out of the production of final output to produce future human capital.
Having in mind the welfare theorems, the representative households problem is
\[ \begin{aligned} & \max _{u, C} U=\int_{0}^{\infty} \frac{C^{1-\theta}-1}{1-\theta} e^{-\rho t} d t \\ & \text { s.t. } \\ & \dot{K}=A K^{\alpha}(u H)^{1-\alpha}-\delta_{K} K-C, \\ & \dot{H}=B(1-u) H-\delta_{H} H \\ & K(0), H(0)>0\text { given. } \end{aligned} \]
Let \(\lambda_{K}\) and \(\lambda_{H}\) be the shadow prices for the physical and human capital, respectively. The problem, if written in terms of current value Hamiltonian, is given by
\[ \max _{u, C} H_{L C}=\frac{C^{1-\theta}-1}{1-\theta}+q_{K}\left[A K^{\alpha}(u H)^{1-\alpha}-\delta_{K} K-C\right]+q_{H}\left[B(1-u) H-\delta_{H} H\right] \]
The optimal rules are
\[ \begin{aligned} & \ C^{-\theta} & =& \lambda_{K} \text {, } \\ & \lambda_{K}(1-\alpha) \frac{Y}{u} & =& \lambda_{H} B H, \\ & \dot{\lambda}_{K}& = & \rho \lambda_{K}-\lambda_{K}\left(\alpha \frac{Y}{K}-\delta_{K}\right) \\ & &= & -\lambda_{K}\left(\alpha \frac{Y}{K}-\delta_{K}-\rho\right), \\ & \dot{\lambda}_{H} & = & \rho \lambda_{H} -\left\{\lambda_{K}(1-\alpha) \frac{Y}{H}+\lambda_{H}\left[B(1-u)-\delta_{H}\right]\right\} . \end{aligned} \tag{9.1}\]
The standard transversality conditions apply for each of the state variables:
\[ \begin{aligned} & \lim _{t \rightarrow \infty} e^{-\rho t} \lambda_{K}(t) K(t)=0 \\ & \lim _{t \rightarrow \infty} e^{-\rho t} \lambda_{H}(t) H(t)=0 \end{aligned} \]
Using the first and third equations in (9.1), it follows that
\[ \frac{\dot{C}}{C}=\frac{1}{\theta}\left(\alpha \frac{Y}{K}-\delta_{K}-\rho\right) \tag{9.2}\]
While from the second and fourth equations, we get:
\[ \begin{aligned} \dot{\lambda}_{H} & = \rho\lambda_{H} -\left\{\lambda_{H} \frac{B H}{(1-\alpha) \frac{Y}{u}}(1-\alpha) \frac{Y}{H}+\lambda_{H}\left[B(1-u)-\delta_{H}\right]\right\} \\ & =\rho \lambda_{H} -\left\{\lambda_{H} B u+\lambda_{H}\left[B(1-u)-\delta_{H}\right]\right\} \Rightarrow \\ \dot{\lambda}_{H} & =-\lambda_{H}\left(B-\delta_{H}-\rho\right) \end{aligned} \]
9.3 Equilibrium and Balanced Growth Path
From the optimal consumption path (10.8) and that the growth rate of consumption at steady state should be constant, it follows that the aggregate output \(Y\) and capital stock \(K\) grow at the same rate, i.e., \(g_{K}=g_{Y}\). From the resource constraint (or the law of motion of capital) \(\frac{\dot{K}}{K}=\) \(\frac{A K^{\alpha}(u H)^{1-\alpha}}{K}-\delta_{K}-\frac{C}{K}=\frac{Y}{K}-\delta_{K}-\frac{C}{K}\) follows that in steady-state the consumption and capital grow at the same rate, i.e., \(g_{C}=g_{K}=g_{Y}\). From the production of human capital, given that \(B\) and \(\delta_{H}\) are constant parameters and in steady-state \(\frac{\dot{H}}{H}=\) is constant, it follows that the share of human capital in production of final good is constant, i.e.,
From the production of final good \(Y=A K^{\alpha}(u H)^{1-\alpha}\) follows that
\[ \frac{Y}{K}=\frac{A K^{\alpha}(u H)^{1-\alpha}}{K}=A\left(u \frac{H}{K}\right)^{1-\alpha} \]
Given that \(g_{K}=g_{Y}\) and \(A, u=\) const the growth rates of physical and human capital are equal, i.e., \(g_{K}=g_{H}=g_{C}=g_{Y} \equiv g\). From (31) and given that \(g_{H}=g_{Y}\) and \(u, \alpha, B=\) constant follows that
\[ \frac{\dot{\lambda}_{H}}{\lambda_{H}}=\frac{\dot{\lambda}_{K}}{\lambda_{K}} \]
This result should not be a surprising result. Given that both human and physical capital should be accumulated in equilibrium, the rates of return on their accumulation \(\frac{\dot{\lambda}_{i}}{\lambda_{i}}(i=H, K)\) should be equal. Otherwise, one of these assets will not be accumulated.
This equality implies then that
\[ -\frac{\dot{\lambda}_{H}}{\lambda_{H}}=\left(B-\delta_{H}-\rho\right)=\left(\alpha \frac{Y}{K}-\delta_{K}-\rho\right)=-\frac{\dot{\lambda}_{K}}{\lambda_{K}} \]
Therefore, from the optimal consumption path (10.8), it follows that
\[ g=\frac{1}{\theta}\left(\alpha \frac{Y}{K}-\delta_{K}-\rho\right)=\frac{1}{\theta}\left(B-\delta_{H}-\rho\right) \tag{9.4}\]
Thus, given that \(g_{H}=g_{C}=g\), from (9.3) it follows that
\[ \begin{aligned} u^{*} & =1-\frac{\left(B-\delta_{H}-\rho\right)+\theta \delta_{H}}{\theta B} \\ & =\frac{(\theta-1)\left(B-\delta_{H}\right)+\rho}{\theta B} \end{aligned} \]
In order to show that \(u^{*}>0\), consider, for instance, the transversality condition for human capital
\[ \lim _{t \rightarrow \infty} e^{-\rho t} \lambda_{H}(t) H(t)=0 \]
Given that in steady-state \(\frac{\dot{\lambda}_{H}}{\lambda_{H}}=-\left(B-\delta_{H}-\rho\right)\) and \(g_{H}=\frac{1}{\theta}\left(B-\delta_{H}-\rho\right) \Rightarrow\) in order the transversality condition to hold
\[ \begin{aligned} -\left(B-\delta_{H}-\rho\right)-\rho+\frac{1}{\theta}\left(B-\delta_{H}-\rho\right) & <0 \Rightarrow \\ \frac{1}{\theta}\left(B-\delta_{H}-\rho\right) & <\left(B-\delta_{H}\right) \Rightarrow \\ (\theta-1)\left(B-\delta_{H}\right)+\rho & >0 \Rightarrow \\ u^{*} & >0 . \end{aligned} \]
Meanwhile, from (9.4) follows that in steady-state
\[ \left(\frac{Y}{K}\right)^{*}=\frac{B-\delta_{H}+\delta_{K}}{\alpha} \]
From the law of motion of capital follows that in steady-state
\[ \begin{aligned} \left(\frac{C}{K}\right)^{*} & =\left(\frac{Y}{K}\right)^{*}-\delta_{K}-\frac{1}{\theta}\left[\alpha\left(\frac{Y}{K}\right)^{*}-\delta_{K}-\rho\right] \\ & =\frac{1}{\theta}\left[(\theta-\alpha) \frac{B-\delta_{H}+\delta_{K}}{\alpha}-(\theta-1) \delta_{K}+\rho\right] \end{aligned} \]
Therefore the savings rate is
\[ s^{*}=\left(\frac{Y-C}{Y}\right)^{*}=\frac{\left(\frac{Y}{K}\right)^{*}-\left(\frac{C}{K}\right)^{*}}{\left(\frac{Y}{K}\right)^{*}}=\frac{\alpha\left(B-\delta_{H}-\rho\right)+\alpha \theta \delta_{K}}{\theta\left(B-\delta_{H}+\delta_{K}\right)} \]
9.4 Comparative statics
- Increase in \(B\) increases \(g\). Ambiguous effects on \(s^{*}\) and \(u^{*}\) (for \(\frac{1}{\theta} \geq 1, s^{*}\) increases and \(u^{*}\) decreases)
- Increase in \(\theta\) (or \(\rho\) ) decreases both \(g\) and \(s^{*}\), while it increases \(u^{*}\)
- Increase in \(\alpha\) increases \(s^{*}\) but has no effect on \(g\) and \(u^{*}\)
9.5 Kaldor stylized facts and first models of endogenous growth
Assume that the aggregate human capital is uniformly distributed across the population: \(H=h L\) and there is no population growth. This suggests that the production function may be thought as one with capital and labour. Human capital plays the role of labour-augmenting technological progress that is endogenously generated by savings from the final output production, i.e. \(Y=A K^{\alpha}(u h L)^{1-\alpha}\). In equilibrium:
\(g=\frac{\dot{H}}{H}=\frac{\dot{h}}{h}\).
\(Y / L=A u^{1-\alpha}\left(\frac{K}{h L}\right)^{\alpha} h\) increases at a rate \(g\) - \(K / L\) also increases at a rate \(g\) - \(Y / K\) is constant
The real interest rate \(r=B-\delta_{H}\) is constant
The wage rate \(w=\frac{\partial Y}{\partial(u L)}=(1-\alpha) \frac{Y}{H} h\) increases at rate \(g\)
Growth rates across countries differ in the long-run due to technology and preference parameters. Initial conditions (initial levels of human and physical capital stock) have a permanent effect on the level of welfare. When economies start with different endowments, the model predicts no convergence in levels of GDP per capita, even if countries have the same long-run growth rate. Parameter changes explain transition dynamics (not covered here) that can accommodate explanations for short episodes of strong growth.
9.6 Further comments
Lucas motivated the importance of human capital accumulation for long-run economic growth, by forming two different (yet complementary) models. The first model allows for human capital accumulation out of the market (e.g., education sector) that would imply that there is a tradeoff between current consumption and future one, since human capital needs to be driven out of current production sector. Furthermore, in his original specification he allowed for both internal and external returns to human capital in the final-good production (spillovers à la Romer, \(Y=\) \(\left.\left(A H^{\gamma}\right) K^{\alpha} H_{Y}^{1-\alpha} ; \gamma>0\right)\).
The second model allows on-the-job accumulation of human capital, i.e., another form of “learning-by-doing”. He assumed multiple goods, with different rates of human capital accumulation as byproduct of their production. The trade-off in this case is that human capital accumulation takes a form of a less desirable mix of current consumption goods. Growth promoting policies implied by either model are very different (education subsidies vs. industrial policy).
The research challenge that Lucas acknowledges himself is that human capital is not a measurable factor, and in particular its potential external effects. He proposes that a good example of the importance of external effects of human capital is the formation of cities.
Overall, the model lacks a good justification of the non-decreasing returns to the human capital accumulation. The accumulation of human capital differs importantly from the accumulation of knowledge and therefore this model is not a model of technological progress. The important difference between them is that human capital is rival and excludable while knowledge is not rival, though can be excludable.
Empirically, human capital growth cannot explain cross-country growth differences. There is only some limited support that human capital matters as an input to R&D. The latter comes out an important factor in driving aggregate productivity and explaining the cross-country variation in the growth and levels of GDP per capita.